#6543933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,6)(3,7,4,8)(17,18,22,27)(19,28,30,40)(20,32,35,42)(21,25,37,26)(23,36,39,31)(24,33,41,34)(29,38,44,47)(43,45,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,36)(22,30)(24,38)(25,28)(26,40)(27,39)(31,37)(32,43)(33,46)(34,45)(35,44)(41,47)(42,48);(10,13)(14,16)(17,20,22,35)(18,24,27,41)(19,29,30,44)(21,33,37,34)(23,38,39,47)(25,32,26,42)(28,43,40,48)(31,45,36,46);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,30)(23,39)(28,40)(29,44)(31,36)(38,47)(43,48)(45,46);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,22,37)(18,25,27,26)(19,31,30,36)(20,33,35,34)(23,40,39,28)(24,32,41,42)(29,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451129);